From UK and Europe

China has many international airports; the most important air gateways are Beijing and Hong Kong. There are at least four airlines fly from London every day into these two cities.

Four or five flights arrive to Hong Kong daily. As well, several other airlines offer indirect services requiring a change of plane en route. Although no other Chinese cities are served directly from Britain, it is possible to reach Shanghai, Macau, Guangzhou, Xiamen and Ürümqi by air via other European or Asian cities. The non-stop flying time from London to Beijing is 10 hours, and 12 to Hong Kong.

There is a daily flight to Hong Kong from Manchester. There are no direct connections from Ireland to China and you will need to stopover in Europe en route.
Finally, if you are using a European airline, you should be able to fly direct to the European hub from the major regional airports rather than traveling to London first.
British Airways (www.britishairways.com) flies to Beijing and Hong Kong.
Virgin Airlines to Shanghai and Hong Kong (www.virgin-atlantic.com).
Cathay Pacific (www.cathaypacific.com) flies directly to Hong Kong.
The prices of KLM Royal Dutch Airlines (www.klm.com) via Amsterdam, Lufthansa (www.lufthansa.com) via Frankfurt, and Finnair (www.finnair.com) via Helsinki, usually are significantly lower.
The prices of eastern European airlines like Tarom Romanian Air Transport (www.tarom.ru) via Bucharest, and with Aeroflot (www.aeroflot.com) via Moscow, or with Asian airlines such as Pakistan International Airlines (www.piac.com.hk) via Islamabad or Karachi, Malaysia Airlines (www.mas.com.my) via Kuala Lumpur, or Singapore Airlines (www.singaporeair.com) via Singapore, might be even cheaper.
There are also some creative route possibilities such as via Ethiopia or the Gulf States.